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I am not sure what $\ A_1\setminus A_0$ equals to... if $A_0=\emptyset$.

I suspect that $\ A_1\setminus A_0$ equals $A_1$. The fact that $A_0=\emptyset$ makes me reason that $\ A_1\setminus\emptyset$ implies $\ A_1\setminus A_0=A_1$ because $\ A_1\setminus\emptyset$ equals "$\ A_1$ without the $\emptyset$". Which is impossible because $\ A_1\setminus\emptyset$ with always consist of $\emptyset$ (general property of empty sets) hence $\ A_1\setminus A_0=A_1$.

Is my reasoning correct? (in addition forgive me in advance for misconceptions of treating the empty set as an element or subset.......)

Post is linked to: Events $A_n\uparrow A$ meaning. $A_n\downarrow A$ meaning.

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For any set $A$ we have $A\setminus\varnothing=A$, so it is indeed the case that $A_1\setminus A_0=A_1$. Recall that $A\setminus B$ is by definition $\{x\in A:x\notin B\}$, so $A\setminus\varnothing=\{x\in A:x\notin\varnothing\}$. And since $x\notin\varnothing$ is true for every $x$, and hence in particular for every $x\in A$, it must be that $A\setminus\varnothing=A$.

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Yes, you are right. In fact

$$A\setminus B =\{x\in A: x\notin B\}$$

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